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IntroductionCollective excitations

Nonlinear featuresConclusions

Nonlinear BEC Dynamics by HarmonicModulation of s-wave Scattering Length∗

Ivana Vidanovic1, A. Balaz1, H. Al-Jibbouri2, A. Pelster3,4

1Scientific Computing Laboratory, Institute of Physics Belgrade, Serbia

2Institut fur Theoretische Physik, Freie Universitat Berlin, Germany

3Faculty of Physik, University Bielefeld, Germany

4Fachbereich Physik, Universitat Duisburg-Essen, Germany

∗Supported by Serbian Ministry of Education and Science (ON171017, NAD-BEC),

DAAD - German Academic and Exchange Service (NAD-BEC),

and European Commission (EGI-InSPIRE, PRACE-1IP and HP-SEE).

Ilija M. Kolarac Foundation, Belgrade April 21, 2011



Overview

Introduction

Experiments with ultracold atomsMean-field description

Collective modes

Excitation of collective modesGaussian approximationLinear vs. nonlinear response

Nonlinear features

Condensate dynamicsExcitation spectraAnalytic perturbative approach

Conclusions

Ilija M. Kolarac Foundation, Belgrade April 21, 2011Vidanovic et al.: Nonlinear BEC Dynamics by Harmonic Modulation of Scattering Length



Experiments with ultracold atomsTheoretical background

Experiments with ultracold atoms

Nobel prize for physics in 2001 for the experimentalachievement of BECCold alkali atoms:Rb, Na, Li, K . . .T ∼ 1nK, ρ ∼ 1014cm−3

Cold bosons, cold fermionsHarmonic trap, optical latticeShort-range interactions,long-range dipolar interactions

Tunable quantum systems concerning dimensionality, typeand strength of interactions




Experiments with ultracold atomsTheoretical background

Mean-field description of a BEC

BEC ⇒ all atoms occupy the same state: ψ(~r, t) is acondensate wave-functionGross-Pitaevskii equation assuming T = 0(no thermal excitations)

i~∂ψ(~r, t)∂t

=[− ~2

2m∆ + V (~r) + g|ψ(~r, t))|2

]ψ(~r, t)

V (~r) = 12mω

2ρ(ρ

2 + λ2z2) is a harmonic trap potentialeffective interaction between atoms is given by g × δ(~r)g = 4π~2Na

m , a is s-wave scattering length, N is number ofatoms in the condensate





BEC with modulated interaction

Usually collective modes are excited by modulation of theexternal trap potentialAn alternative way of excitation - recent experiment byHulet’s and Bagnato’s group: PRA 81, 053627 (2010)

BEC of 7Li is confined in acylindrical trap

Time-dependent modulation ofatomic interactionsvia a Feshbach resonance

Interesting setup for studyingnonlinear BEC dynamics

300 µm





Gaussian approximation

To simplify calculations and to obtain analytical insight,we approximate density of atoms by a Gaussian variationalansatzFor a spherically symmetric trap

ψG(r, t) = N (t) exp

»−1

2

r2

u(t)2+ ir2φ(t)

–By extremizing corresponding action, we obtain anordinary differential equation, PRL 77, 5320 (1996)In the dimensionless form

u(t) + u(t)− 1

u(t)3− p(t)

u(t)4= 0

Interaction: p(t) =q

2πNa(t)/l, l =

p~/mωρ





Linear response

Using this type of approximation and relying on the linearstability analysis, frequencies of low-lying collective modeshave been analytically calculatedThe equilibrium width

u0 =1

u30

+p

u40

Linear stability analysis

u(t) = u0 + δu(t)⇒ δu+ ω20δu = 0

ω0 =

s1 +

3

u40

+4p

u50





Beyond linear response - motivation

Due to the nonlinear form of the underlying GP equation,we have nonlinearity induced shifts in the frequencies oflow-lying modes (beyond linear response)Our aim is to describe collective modes induced byharmonic modulation of interaction

p(t) ' p+ q cos Ωt

q - modulation amplitude, Ω - modulation frequencyFor Ω close to some BEC eigenmode we expect resonances- large amplitude oscillations and role of nonlinear termsbecomes crucial




Condensate dynamicsExcitation spectraAnalytic perturbative approachResults

Condensate dynamics

u(t) + u(t)− 1u(t)3

− p

u(t)4− q

u(t)4cos Ωt = 0

p = 0.4, q = 0.1, u(0) = u0, u(0) = 0, ω0 = 2.06638

Dynamics depends on Ω

1.02 1.04 1.06 1.08 1.1

1.12 1.14

0 5 10 15 20 25 30 35 40

u(t)

t

= 1, analytics = 1, numerics

0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 5 10 15 20 25 30 35 40t


0 1 2 3 4 5 6

0 50 100 150 200 250 300 350 400t

= 2.04, numerics

1.07

1.08

1.09

1.1

1.11

0 5 10 15 20

u(t)

t


0 0.5

1 1.5

2 2.5

3 3.5

4 4.5

0 50 100 150 200 250 300 350t

= 4.1, numerics

1.08

1.09

0 5 10 15 20t






Excitation spectra (1)

We look at the Fourier transform of u(t),p = 0.4, q = 0.1 and Ω = 2

10-610-510-410-310-210-1100101102103

-40 -30 -20 -10 0 10 20 30 40Frequency

= 2

10-210-1100101102

0.05 0.06 0.07 0.08Frequency

-

10-210-1100101102

1.9 2 2.1 2.2Frequency

10-410-2100102

4 4.05 4.1 4.15 4.2Frequency

2 +

2

10-310-210-1100101

6 6.05 6.1 6.15 6.2Frequency

3

2 + + 23





Excitation spectra (2)

Frequency of the breathing mode is significantly shifted inthe resonant region

10-2

10-1

100

101

1.98 2.02 2.06 2.1Frequency

0

= 1

10-210-1100101102

1.98 2.02 2.06 2.1 2.14Frequency

0 = 2





Analytic perturbative approach (1)

Linear stability analysis yields zeroth order collective modeω0 of oscillations around the time-independent solution u0:

u0 −1

u30

− p

u40

= 0, ω0 =

s1 +

3

u40

+4p

u50

To calculate the collective mode to higher orders, werescale time as s = ωt:

ω2 u(s) + u(s)− 1

u(s)3− p

u(s)4− q

u(s)4cos

Ωs

ω= 0

We assume the following perturbative expansions in q:

u(s) = u0 + q u1(s) + q2 u2(s) + q3 u3(s) + . . .

ω = ω0 + q ω1 + q2 ω2 + q3 ω3 + . . .






This leads to a hierarchical system of equations:

ω20 u1(s) + ω2

0 u1(s) =1

u40

cosΩs

ω

ω20 u2(s) + ω2

0 u2(s) = −2ω0 ω1 u1(s)− 4

u50

u1(s) cosΩs

ω+ αu1(s)2

ω20 u3(s) + ω2

0 u3(s) = −2ω0 ω2 u1(s)− 2β u1(s)3 + 2αu1(s)u2(s)− ω21 u1(s)

+10

u60

u1(s)2 cosΩs

ω− 4

u50

u2(s) cosΩs

ω− 2ω0 ω1 u2(s)

where α = 10p/u60 + 6/u5

0 and β = 10p/u70 + 5/u6

0.

We determine ω1 and ω2 by imposing cancellation ofsecular terms - Poincare-Lindstedt method





Results

Frequency of the breathing mode vs. driving frequency ΩResult in the second order of the perturbation theory

ω = ω0 + q2 Polynomial(Ω)(Ω2 − ω2

0)2 (Ω2 − 4ω20)

+ . . .

2

2.02

2.04

2.06

2.08

2.1

1 1.5 2 2.5 3 3.5 4 4.5 5

Freq

uenc

y

0analytical numerical

1.9 1.95

2 2.05 2.1

2.15 2.2

2.25 2.3

1 2 3 4 5 6

Freq

uenc

y

0analytical numerical

p = 0.4, q = 0.1 p = 1, q = 0.8





Experimental setup - results

p = 15, q = 10, λ = 0.021,ωQ0 = 2π × 8.2 Hz, ωB0 = 2π × 462 Hz

ωB >> ωQ, Ω ∈ (0, 3ωQ),large modulationamplitudeStrong excitation ofquadrupole modeExcitation of breathingmode in the radialdirectionFrequency shifts ofquadrupole mode of about10% are present

0.03

0.032

0.034

0.036

0.038

0.04

0 0.02 0.04 0.06 0.08 0.1 0.12

Freq

uenc

y Q

Q0Q, (a)Q, (n)




Conclusions

Motivated by recent experimental results, we have studiednonlinear BEC dynamics induced by harmonicallymodulated interactionWe have used a combination of an analytic perturbativeapproach, numerics based on Gaussian approximation andnumerics based on full time-dependent GP equationRelevant excitation spectra have been presented andprominent nonlinear features have been found: modecoupling, higher harmonics generation and significant shiftsin the frequencies of collective modesOur results are relevant for future experimental designsthat will include mixtures of cold gases and theirdynamical response to harmonically modulated interactions





Secular term - explanation

x(t) + ω2x(t) + C cos(ωt) = 0

x(t) = A cos(ωt) +B sin(ωt)− C

2ωt sin(ωt)︸︷︷︸

linear in t

In order to have properly behaved perturbative expansion,we impose cancellation of secular terms by appropriatelyadjusting ω1 and ω2

Another way of reasoning

u(t) = A cosωt+A1t sinωt ≈ A cosωt cos ∆ωt+A1

∆ωsin ∆ωt sinωt

u(t) ≈ A cos[(ω −∆ω)t]⇒ ∆ω = A1/A


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